Combinatorial addition formulas and applications (Q696807)

This paper develops analogues of addition formulae such as \(\exp(x+y)=\exp(x)\exp(y)\), but using combinatorial species instead of, in the example just cited, the exponential function. Here a combinatorial species, in the sense of Joyal, is a class of finite labelled structures which is closed under relabelling along bijections. Any combinatorial species can be written as a sum of irreducible ``molecular'' species. The main results are abstract generalizations of multinomial expansions and colouring for molecular species. They are followed by a number of more concrete examples dealing with lists, chains, sets, cycles and palindromes. Further applications are given to the calculation of generalized binomial coefficients (as introduced by the authors in [J. Comb. Theory, Ser. A 91, 15-48 (2000; Zbl 0956.05102)]) and to cycle index series and asymmetry index series.